1.17.6: Isotonic Method; Isopiestic Method

The isotonic method is ‘beautifully simple’[1]. The technique, described as both Isotonic and Isopiestic, leads to osmotic coefficients for solvents and activity coefficients for solutes in solution, generally aqueous solutions. Authors reporting their results describe apparatus and procedures which often differ marginally from those of other authors. Scatchard and coworkers describe how six small platinum cups, volume approx. \(15 \mathrm{~cm}^{3}\), are held in a gold-plated copper block, the cups being fitted with hinged lids [1]. The cups and copper block, filled with solutions (see below) are held in a partially evacuated thermostatted chamber. The copper block is rocked gently. Over a period of time the cups are removed, weighed and replaced.The experiment ends when the masses of solutions in the cups are constant.

The development of the isopiestic method can be traced to the experiments reported in 1917 by Bousfield[2] (who used the word, iso-piestic). A closed system was set up containing several solid salts in separate sample cells together with a little water(\(\ell\)) in a separate sample cell. A little more water(\(\ell\)) was added to the separate sample cell and the sample cells containing salts reweighed over a period of many days. The uptake of water by the salts was monitored, eventually forming salt solutions. A quantity \(\mathrm{h}\), the number of moles of water taken up by a mole of salt was calculated; e.g. \(\mathrm{h } = 12.43(\mathrm{KCl}), 14.23 (\mathrm{NaCl}) \text { and } 17.18(\mathrm{LiCl})\). The system is isopiestic, meaning that all samples have equal vapour pressure. [One cannot help but feel sorry for Bousfield after reading the Discussion after the paper was presented at a meeting. The critics clearly did not appreciate what Bousfield was attempting to do.] Modern techniques developed from this approach[3].

To illustrate the technique, consider the case where just two cups, \(\mathrm{A}\) and \(\mathrm{B}\), are used containing aqueous solutions of two salts, \(\mathrm{i}\) and \(\mathrm{j}\). Spontaneous transfer of solvent water occurs through the vapour phase until eventually (often after many hours) equilibrium is attained and no change in mass occurs [4,5]. At equilibrium the chemical potentials of water in the two dishes are equal. Thus,

\[\mu_{\mathrm{j}}^{\mathrm{eq}}(\operatorname{dish} \mathrm{A}, \mathrm{T}, \mathrm{p})=\mu_{\mathrm{i}}^{\mathrm{cq}}(\operatorname{dish} \mathrm{B}, \mathrm{T}, \mathrm{p})\]

Granted that the masses of salts used to prepare the solutions in the two cups are accurately known, the mass of cups at equilibrium yields the equilibrium molalities. In most studies one dish (e.g. dish \(\mathrm{A}\)) holds a standard [e.g. \(\mathrm{KCl}(\mathrm{aq})\)] for which the dependence of practical osmotic coefficient on composition is accurately known.

If, for example, the two cups contain aqueous salt solutions, equation (a) is rewritten as follows granted that the pressure is close to ambient.

\[\left[\mu_{1}^{*}(\ell)-\phi_{\mathrm{j}} \, \mathrm{R} \, \mathrm{T} \, \mathrm{v}_{\mathrm{j}} \, \mathrm{m}_{\mathrm{j}}\right]_{\mathrm{A}}=\left[\mu_{1}^{*}(\ell)-\phi_{\mathrm{i}} \, \mathrm{R} \, \mathrm{T} \, \mathrm{v}_{\mathrm{i}} \, \mathrm{m}_{\mathrm{i}}\right]_{\mathrm{B}}\]

Here \(\mathrm{m}_{\mathrm{i}}\) and \(\mathrm{m}_{\mathrm{j}}\) are the equilibrium molalities, where the word ‘equilibrium’ refers to the solvent water. Hence

\[\left(\phi_{\mathrm{j}} \, \mathrm{v}_{\mathrm{j}} \, \mathrm{m}_{\mathrm{j}}\right)_{\mathrm{A}}=\left(\phi_{\mathrm{i}} \, \mathrm{v}_{\mathrm{i}} \, \mathrm{m}_{\mathrm{i}}\right)_{\mathrm{B}}\]

The isopiestic ratio \(\mathrm{R}_{\text{iso}}\) is defined by equation (d).

\[\mathrm{R}_{\text {iso }}=\left(\mathrm{v}_{\mathrm{i}} \, \mathrm{m}_{\mathrm{i}}\right)_{\mathrm{B}} /\left(\mathrm{v}_{\mathrm{j}} \, \mathrm{m}_{\mathrm{j}}\right)_{\mathrm{A}}\]

Hence,

\[\phi_{\mathrm{B}}=\phi_{\mathrm{A}} / \mathrm{R}_{\mathrm{iso}}\]

Therefore \(\phi_{\mathrm{B}}\) is obtained from the experimentally determined \(\mathrm{R}_{\text{iso}}\) and a known (i.e. previously published standard) \(\phi_{\mathrm{B}}\).

In general terms, an ‘isopiestic experiment’ is based around the properties of the solvent water in a given solution. But the aim of the experiment is to gain information about the activity coefficient of the solute. The calculation therefore relies on the Gibbs - Duhem Equation . According to the Gibbs – Duhem equation the dependence of chemical potentials of salt and solvent are linked . If a given solution comprises \(\mathrm{n}_{1}\) moles of water and \(\mathrm{n}_{\mathrm{j}}\) moles of solute, then

\[\mathrm{n}_{1} \,\left(\mathrm{d} \mu_{1} / \mathrm{dn} \mathrm{n}_{\mathrm{j}}\right)+\mathrm{n}_{\mathrm{j}} \,\left(\mathrm{d} \mu_{\mathrm{j}} / \mathrm{dn}_{\mathrm{j}}\right)=0\]

For a solution molality \(\mathrm{m}_{\mathrm{j}}\) in a solvent, molar mass \(\mathrm{M}_{1}\)

\[\left(1 / \mathrm{M}_{1}\right) \, \mathrm{d} \mu_{\mathrm{1}} / \mathrm{dm} \mathrm{m}_{\mathrm{j}}+\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \mu_{\mathrm{j}} / \mathrm{dm}_{\mathrm{j}}=0\]

Then if pressure \(\mathrm{p}\) is close to the standard pressure,

\[\left(1 / \mathrm{M}_{1}\right) \, \mathrm{d}\left[\mu_{1}^{*}(\ell)-\phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}\right] / \mathrm{dm}_{\mathrm{j}}+\mathrm{m}_{\mathrm{j}} \, \mathrm{d}\left[\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)\right] / \mathrm{dm}_{\mathrm{j}}=0\]

Or,

\[-\phi-\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \phi / \mathrm{dm}_{\mathrm{j}}+1+\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ln \left(\gamma_{\mathrm{j}}\right) / \mathrm{dm}_{\mathrm{j}}=0\]

Thus

\[-\left(\phi / \mathrm{m}_{\mathrm{j}}\right)-\mathrm{d} \phi / \mathrm{dm} \mathrm{m}_{\mathrm{j}}+\left(1 / \mathrm{m}_{\mathrm{j}}\right)+\mathrm{d} \ln \left(\gamma_{\mathrm{j}}\right) / \mathrm{dm} \mathrm{m}_{\mathrm{j}}=0\]

No further progress can be made until we have determined in a series of experiments the dependence of \(\phi\) on \(\mathrm{m}_{\mathrm{j}}\). The dependence of \(\phi_{\mathrm{B}}\) on molality \(\mathrm{m}_{\mathrm{B}}\) is obtained after many experiments. In a common procedure the dependence is fitted to a polynomial in mj such that integration yields the activity coefficient for the solute \(\gamma_{\mathrm{j}}\). Suppose for example we find that for a given system \(\phi\) is a linear function of molality \(\mathrm{m}_{\mathrm{j}}\). Thus

\[\phi=1+\mathrm{a} \, \mathrm{m}_{\mathrm{j}}\]

Or,

\[\mathrm{d} \phi / \mathrm{dm}_{\mathrm{j}}=\mathrm{a}\]

Hence,

\[\ln \left(\gamma_{\mathrm{j}}\right)=2 \, \mathrm{a} \, \mathrm{m}_{\mathrm{j}}\]

We note how the analysis relies on the fact the solute and solvent ‘communicate with each other’.

Footnotes

[1] G. Scatchard, W.J.Hamer and S.E.Wood, J.Am.Chem.Soc.,1938,60,3061.

[2] W. R. Bousfield, Trans. Farady Soc.,1917,13,401.

[3] J. A.Rard and R. F. Platford, Activity Coefficients in Electrolyte Solutions, ed. K. S. Pitzer, CRC Press, Boca Raton, 2nd edition, 1991.

[4] In fact one can regard the phenomenon as osmosis, the vapour phase being a perfect semi-permeable membrane.

[5] The system is partially evacuated so that equilibrium vapour pressure is reasonably rapidly attained.